PROPOSTA DE UM TESTE DE NORMALIDADE MULTIVARIADA EXATO BASEADO EM UMA TRANSFORMAÇÃO t DE STUDENT

Abstract

The normal distribution is one of the most important continuous probability distribution. This distribution describe several phenomena and plays an important role in inferential statistics. It is well known that the normality directly infuences the quality and reliability of scientic research, since violations of normality assumption can lead to incorrect results and conclusions. The same is expected for multivariate normal distribution inferences. A simple manner, however subjective, to verify the univariate or multivariate normality is through quantile-quantile plots (Q-Q plots). Furthermore, the Q-Q plots are eficient tools for visualization of outliers. The drawback of the classical Q-Q plot is that the Mahalanobis distance quantiles are only asymptotically identical, but not independently, chi-squared distributed. This fact compromises the eficiency of the Q-Q plot or any test based on the use of the observed distance quantiles. The aim of this study is to propose an accurate test and validate its performance by Monte Carlo simulation and also provide a Q-Q plot to detect further evidence of violation of multivariate normality in p dimensions. This Q-Q plot originates from a characterization of the multivariate normal distribution made by Yang et al. (1996) based on the spherical distribution properties (Fang et al., 1990). The R program version 3:1:0 was used to implement this Q-Q plot normality test and to verify its performance by Monte Carlo simulations. The Monte Carlo simulation showed that the proposed test successful controls the type I error rates being accurate, but shows lower power than any other multivariate normality test.